OF LOCAL THEOREMS AND POR1SMS. 347 



but this does not satisfy the condition of rendering the curve 

 symmetrical relative to its axis. 



Let a. x zz s/a 2 — x 2 , then we have 



__ V c 2 — a 2 + 2x v/fl 2 — x 2 . 9 x 

 y — _ j 



9X + 9 \/a 2 — x 2 



and the following property belongs to all the curves contained 

 in this family : 



Any curve BAC of this species being given. Fig. 7., a right line 

 BC given in length may be found such, that if it be any how pla- 

 ced, so that its extremities B and C coincide with any two points of 

 similar branches, but on opposite sides of the axis ; then the sum 

 of the squares described on the abscissce AE, AD, shall be equal 

 to the square described on a right line which may be found. 



If we make u x zz — x, we have the family of curves ex- 

 pressed by the equation 



\/c 2 — 4 x 2 . 9 x 

 y ~~ 9 x -f- 9 ( — x) 



which possess the following property : 



Any one of this class of curves being given, a right line CB may 

 be found, which, if it be placed so that its extremities rest on two 

 opposite branches of the curve (at C and B), and if the ordinate 

 at one of these points be prolonged on the other side the axis, un- 

 til the part produced (FE) be equal to the abscissa (GH) at the 

 other point, the extremity of the line so produced (E), will be si- 

 tuated in a straight line given by position. 



If we make ? x zz b, the equation becomes y zz J- , x * 



and the curve is a circle whose radius is -, the line to be 



found 



